[Merged by Bors] - feat(category_theory): left-derived functors #7487
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…ntially surjective
Co-authored-by: Johan Commelin <johan@commelin.net>
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# Left-derived functors We define the left-derived functors `F.left_derived n : C ⥤ D` for any additive functor `F` out of a category with projective resolutions. The definition is ``` projective_resolutions C ⋙ F.map_homotopy_category _ ⋙ homotopy_category.homology_functor D _ n ``` that is, we pick a projective resolution (thought of as an object of the homotopy category), we apply `F` objectwise, and compute `n`-th homology. We show that these left-derived functors can be calculated on objects using any choice of projective resolution, and on morphisms by any choice of lift to a chain map between chosen projective resolutions. Similarly we define natural transformations between left-derived functors coming from natural transformations between the original additive functors, and show how to compute the components. ## Implementation We don't assume the categories involved are abelian (just preadditive, and have equalizers, cokernels, and image maps), or that the functors are right exact. None of these assumptions are needed yet. It is often convenient, of course, to work with `[abelian C] [enough_projectives C] [abelian D]` which (assuming the results from `category_theory.abelian.projective`) are enough to provide all the typeclass hypotheses assumed here. Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Johan Commelin <johan@commelin.net>
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# Left-derived functors We define the left-derived functors `F.left_derived n : C ⥤ D` for any additive functor `F` out of a category with projective resolutions. The definition is ``` projective_resolutions C ⋙ F.map_homotopy_category _ ⋙ homotopy_category.homology_functor D _ n ``` that is, we pick a projective resolution (thought of as an object of the homotopy category), we apply `F` objectwise, and compute `n`-th homology. We show that these left-derived functors can be calculated on objects using any choice of projective resolution, and on morphisms by any choice of lift to a chain map between chosen projective resolutions. Similarly we define natural transformations between left-derived functors coming from natural transformations between the original additive functors, and show how to compute the components. ## Implementation We don't assume the categories involved are abelian (just preadditive, and have equalizers, cokernels, and image maps), or that the functors are right exact. None of these assumptions are needed yet. It is often convenient, of course, to work with `[abelian C] [enough_projectives C] [abelian D]` which (assuming the results from `category_theory.abelian.projective`) are enough to provide all the typeclass hypotheses assumed here. Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Johan Commelin <johan@commelin.net>
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Left-derived functors
We define the left-derived functors
F.left_derived n : C ⥤ Dfor any additive functorFout of a category with projective resolutions.
The definition is
that is, we pick a projective resolution (thought of as an object of the homotopy category),
we apply
Fobjectwise, and computen-th homology.We show that these left-derived functors can be calculated
on objects using any choice of projective resolution,
and on morphisms by any choice of lift to a chain map between chosen projective resolutions.
Similarly we define natural transformations between left-derived functors coming from
natural transformations between the original additive functors,
and show how to compute the components.
Implementation
We don't assume the categories involved are abelian
(just preadditive, and have equalizers, cokernels, and image maps),
or that the functors are right exact.
None of these assumptions are needed yet.
It is often convenient, of course, to work with
[abelian C] [enough_projectives C] [abelian D]which (assuming the results from
category_theory.abelian.projective) are enough toprovide all the typeclass hypotheses assumed here.